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Polynomial # 1 (Posted on 2006-05-26) Difficulty: 3 of 5
Let f(x) be a nonconstant polynomial in x with integer coefficients and suppose that for five distinct integers a1, a2, a3, a4, a5, one has f(a1)= f(a2)= f(a3)= f(a4)= f(a5)= 2.

Find all integers b such that f(b)= 9.

No Solution Yet Submitted by Ravi Raja    
Rating: 5.0000 (1 votes)

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Solution Solution (spoiler) | Comment 2 of 7 |
If f(x)=2 at a1, a2, a3, a4, and a5, then g(x)=f(x)-2 has those roots, and can be written as (x-a1)(x-a2)(x-a3)(x-a4)(x-a5)h(x), where h(x) is another polynomial with integer coefficients.

If f(x)=9, then g(x)=7...  but for integer x, g(x) is the product of at least 5 different integer values, and we cannot write 7 that way, for any values of a1 thorugh a5.

For the extra question by Federico Kereki, we are looking for a square S such that S-2 can be written as the product of at lest 5 different integer values. I found 100 might do the job, for 98=(-7).(-2).(-1).(1).(7) with h(x)=1, but of course, this would work only for very specific values of a1, a2, a3, a4, and a5: a1=any, a2=a1-5, a3=a2-1, a4=a3-2, a5=a4-6, and then b=a1-7.

  Posted by e.g. on 2006-05-26 12:43:32
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